Here is a familiar puzzle from the Mathigon Puzzle Calendars for 2021.
“Given a line and two points A and B, which point P on the line forms the largest angle APB?”
See the Max Angle Puzzle
An excellent application of the solution to this puzzle can be found at Numberphile, where Ben Sparks explains an optimal rugby goal-kicking strategy.
(Update 3/23/2023) Solution Construction
Oscar Rojas has provided a construction of the solution to the problem:
“An interesting fact about the solution to this problem, is that such point P (where the circle is tangent to the line) can be found by a straightedge and compass construction.”